In this post, I will explain how to built a 4-D torus that would tell why a matter wave wavelength depends on its speed (De Broglie wavelength).
1) Take a slinky. Make a mark along the slinky. I cut out an inch of tape paper and attached it to each spiral. Then, I marked one side of the tape paper, this represent "arrows".
2) Make a clockwise turn around the slinky edge and glue the tail arrow with the head arrow. This makes a torus. The twisted path of the arrows is very noticeable.
3) Take another slinky. Do the same as before but a counter-clockwise turn.
Heuristically, these slinkies have to be together. It is too difficult to see anything when they are together, so I draw them separately. You have to arrange them in the way shown in Figure 1. The clockwise-turned slinky is at the right of the Figure. The movement of its arrows is clockwise. The counterclockwise-turned slinky is at the left of the Figure. The movement of its arrows is counterclockwise. This makes that the wavelength left on flatland depends on the object speed.
These arrows are electric field vectors that get printed in the plane. The impression will look like this:
Figure 2 electric field vectors imprinted in the plane. The wavelength will depend on the speed of the object and will use the two dimensions of the plane to travel.
On Fig 2 undistinguishables states occur at 0, 180 and 360 degrees. Notice that the printed arrows close a circle when the particle arrive to 180 degrees. Thus, an observer in the plane may judge that one complete round has been made. However, the intersection at 180 degrees is different to the original state at 0 degrees. After another round, the printed pattern is exactly like the state at the origin. This is a characteristic of a spin 1/2 particle.